LQR (linear quadratic regulator) is one of the earliest and most mature state space design methods in modern control theory [9]. The optimal control law of state linear feedback is obtained by LQR, and it is easy to achieve closed-loop optimal control [10].

The linear dynamic model of satellite formation flying is rewritten as a state-space expression.
$$\dot{x}(x)=A(x)x+B(x)u$$(10)

In equation(10), *u* represents the system control variables:
$u={\left[\begin{array}{c}{f}_{x}\phantom{\rule{1em}{0ex}}{f}_{y}\phantom{\rule{1em}{0ex}}{f}_{z}\end{array}\right]}^{\mathrm{T}}={\left[\begin{array}{c}{u}_{x}\phantom{\rule{1em}{0ex}}{u}_{y}\phantom{\rule{1em}{0ex}}{u}_{z}\end{array}\right]}^{\mathrm{T}};$

*x* indicates the system state variable:
$$\begin{array}{}x\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}={\left[\begin{array}{c}x\phantom{\rule{1em}{0ex}}y\phantom{\rule{1em}{0ex}}z\phantom{\rule{1em}{0ex}}{V}_{x}\phantom{\rule{1em}{0ex}}{V}_{y}\phantom{\rule{1em}{0ex}}{V}_{z}\end{array}\right]}^{\mathrm{T}}\\ \phantom{\rule{1em}{0ex}}={\left[\begin{array}{c}x\phantom{\rule{1em}{0ex}}y\phantom{\rule{1em}{0ex}}z\phantom{\rule{1em}{0ex}}\dot{x}\phantom{\rule{1em}{0ex}}\dot{y}\phantom{\rule{1em}{0ex}}\dot{z}\end{array}\right]}^{\mathrm{T}};\end{array}$$

*A*(*x*) is the state matrix of the system:
$$\begin{array}{}A(x)=\left[\begin{array}{cc}{0}_{3\times 3}& {A}_{1}\\ {A}_{2}& {A}_{3}\end{array}\right];\end{array}$$

In the matrix,
$\begin{array}{}{A}_{1}=\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right],\end{array}$
$$\begin{array}{}{A}_{2}=\left[\begin{array}{ccc}0& 0& 0\\ 0& 3{n}^{2}& 0\\ 0& 0& -{n}^{2}\end{array}\right],\phantom{\rule{thinmathspace}{0ex}}{A}_{3}=\left[\begin{array}{ccc}0& -2n& 0\\ 2n& 0& 0\\ 0& 0& 0\end{array}\right].\end{array}$$

*B*(*x*) is a control matrix:
$$\begin{array}{}B(x)=\left[\begin{array}{c}{0}_{3\times 3}\\ {B}_{1}\end{array}\right];\end{array}$$

In the matrix,
$\begin{array}{}{B}_{1}=\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right].\end{array}$

Assume the altitude of the leader-satellite is 800 km, the orbital radius of the leader-satellite is *r*_{c} = 7.2 × 10^{5} m, the orbital angular velocity is *n* = 0.045 rad/sec = 2.5 × 10^{−4} rad/sec. At this time the state space expression is:
$$\begin{array}{}\left[\begin{array}{c}\dot{x}\\ \dot{y}\\ \dot{z}\\ \ddot{x}\\ \ddot{y}\\ \ddot{z}\end{array}\right]=D\left[\begin{array}{c}x\\ y\\ z\\ \dot{x}\\ \dot{y}\\ \dot{z}\end{array}\right]+\left[\begin{array}{ccc}0& 0& 0\\ 0& 0& 0\\ 0& 0& 0\\ 1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right]\left[\begin{array}{c}{u}_{x}\\ {u}_{y}\\ {u}_{z}\end{array}\right]\end{array}$$(11)

In equation(11),
$$\begin{array}{}& D=\\ & \left[\begin{array}{cccccc}0& 0& 0& 1& 0& 0\\ 0& 0& 0& 0& 1& 0\\ 0& 0& 0& 0& 0& 1\\ 0& 0& 0& 0& -5\times {10}^{-4}& 0\\ 0& 1.875\times {10}^{-7}& 0& 5\times {10}^{-4}& 0& 0\\ 0& 0& -6.25\times {10}^{-8}& 0& 0& 0\end{array}\right]\\ \\ & \phantom{\rule{1em}{0ex}}rank({W}_{c})=\left[\begin{array}{cccccc}B& AB& {A}^{2}B& {A}^{3}B& {A}^{4}B& {A}^{5}B\end{array}\right]=6\end{array}$$(12)

This indicates that the system can be controlled.

The objective function of LQR theory is:
$${J}_{c}=\frac{1}{2}\underset{{t}_{0}}{\overset{{t}_{f}}{\int}}[(x-{x}_{d}{)}^{\mathrm{T}}Q(x-{x}_{d})+{u}^{\mathrm{T}}Ru]dt$$(13)

In equation(13), *x*_{d} is an ideal state, *Q* is used as the weight matrix of the error in the optimization process, and is a positive definite constant matrix of 6 × 6; *R* is the weight matrix of the control variables in the optimization process and is a positive semi-definite constant matrix of 3 × 3.

Solving Riccati equation:
$$-\dot{P}=P(x)A(x)+{A}^{\mathrm{T}}(x)P(x)-P(x)B(x){R}^{-1}(x)P(x)+Q(x)$$(14)

*P*(*x*) can be solved.

Feedback matrix:
$$K(x)=-{R}^{-1}(x){B}^{\mathrm{T}}(x)P(x)$$(15)

The control law of the linear system in the performance index is:
$$u(x)=-K(x)(x-{x}_{d})$$(16)

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